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Let G be a connected reductive algebraic group defined over an algebraic closure of the finite field of prime order p > 0, which we assume to be good for G. We denote by F : G → G a Frobenius endomorphism of G and by G the corresponding Fq-rational structure. If Irr(G) denotes the set of ordinary irreducible characters of G then by work of Lusztig and Geck we have a well defined map ΦG : Irr(G) → {F-stable unipotent conjugacy classes of G} where ΦG(χ) is the unipotent support of χ. Lusztig has given a classification of the irreducible characters of G and obtained their degrees. In particular he has shown that for each χ ∈ Irr(G) there exists an integer nχ such that nχ · χ(1) is a monic polynomial in q. Given a unipotent class O of G with representative u ∈ G we may define AG(u) to be the finite quotient group CG(u)/CG(u)◦. If the centre Z(G) is connected and G/Z(G) is simple then Lusztig and H´ezard have independently shown that for each F-stable unipotent class O of G there exists χ ∈ Irr(G) such that ΦG(χ) = O and nχ = |AG(u)|, (in particular the map ΦG is surjective). The main result of this thesis extends this result to the case where G is any simple algebraic group, (hence removing the assumption that Z(G) is connected). In particular if G is simple we show that for each F-stable unipotent class O of G there exists χ ∈ Irr(G) such that ΦG(χ) = O and nχ = |AG(u)F| where u ∈ OF is a well-chosen representative. We then apply this result to prove, (for most simple groups), a conjecture of Kawanaka’s on generalised Gelfand–Graev representations (GGGRs). Namely that the GGGRs of G form a Z-basis for the Z-module of all unipotently supported class functions of G. Finally we obtain an expression for a certain fourth root of unity associated to GGGRs in the case where G is a symplectic or special orthogonal group.

This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.

This paper is primarily concerned with the fundamental properties of a linear algebra of finite order over a field. A discussion of linear sets of finite order over a field is used as an introduction to these properties.

Kung Kin Chung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 60-63). / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- 2-D Thermoelastic System --- p.5 / Chapter 2.1 --- Introduction to the thermo-elastic system --- p.5 / Chapter 2.2 --- Existing theoretical results on the system --- p.6 / Chapter 2.2.1 --- Well-posedness and energy decay phenomena of the thermoelastic system --- p.7 / Chapter 2.2.2 --- Reduction of the coupled thermoelastic sys- tem to a single heat equation --- p.8 / Chapter 2.3 --- Some existing numerical methods --- p.12 / Chapter 2.3.1 --- Explicit scheme --- p.12 / Chapter 2.3.2 --- Implicit scheme --- p.16 / Chapter 2.4 --- The formulation and transformation --- p.17 / Chapter 2.5 --- Finite element discretizations --- p.19 / Chapter 2.6 --- Existence of the discrete solution --- p.20 / Chapter 2.7 --- Numerical results --- p.23 / Chapter 2.7.1 --- Introduction to the FEM software --- p.23 / Chapter 2.7.2 --- Numerical experiments --- p.28 / Chapter 3 --- 3-D Elliptic Interface Problem --- p.35 / Chapter 3.1 --- Introduction to 2-D elliptic problems --- p.35 / Chapter 3.2 --- Finite element discretization --- p.36 / Chapter 3.3 --- Convergence of the numerical methods --- p.38 / Chapter 3.4 --- Numerical results --- p.43 / Chapter 3.4.1 --- Introduction --- p.43 / Chapter 3.4.2 --- Introduction to the FEM software --- p.45 / Chapter 3.4.3 --- Numerical experiments --- p.47 / Chapter 4 --- 2-D Parabolic Interface Problem --- p.53 / Chapter 4.1 --- Introduction to the 2-D parabolic problem --- p.53 / Chapter 4.2 --- Finite element discretization --- p.54 / Chapter 4.3 --- Existence and convergence analysis --- p.55 / Chapter 4.4 --- Numerical results --- p.57 / Chapter 4.4.1 --- Introduction --- p.57 / Chapter 4.4.2 --- Numerical experiments --- p.58 / Bibilography

The main portion of my thesis focuses on a 2-dimensional second order heat transmission problem in domains with pre-fractal interfaces. My focus is on the numerical approximation of the solutions. Precisely, I€™m concerned to develop a suitable mesh refinement algorithm that could be adapted to our situation, by taking into account the regularity of the solutions and the geometry of irregular pre-fractal interfaces. I obtain an error estimate between the weak solution and the discrete solution, which indicates an optimal rate of convergence as in the classical case when the solution has H^2-regularity. In addition, numerical simulations are also included, which demonstrates the features of our heat transmission model. Another portion of my thesis focuses on the asymptotic analysis of singular boundary value problems with highly conductive layers of pre-fractal type. My models illustrate the problems of a lower- dimensional highly conductive material intruding into a higher- dimensional material with lower conductivity. I consider a 2D model of Sierpinski pre-fractal layers and 3D models of hierarchical layers. The main results consist in the so-called Mosco-convergence of certain energy functionals, which implies the strong convergence of the solutions and of the spectral resolutions as a byproduct in real applications.

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Cette étude concerne la simulation de systèmes thermiques comportant plusieurs domaines à l’aide d’une méthode d’ordre réduit adaptée au mono-corps. Les modèles modaux réduits sont construits à partir de modes de branche. Ceux-ci sont déterminés à l’aide d’un modèle détaillé et permettent la prise en compte de non-linéarités. La réduction est effectuée par la technique de l’amalgame modal découplé qui permet de façon automatique et immédiate d’obtenir une base réduite qui ne comporte qu’un faible nombre de modes à partir de la base initiale. Le couplage entre les différents domaines s’effectue par l’intermédiaire d’une résistance thermique de contact. On montre la nécessité de faire intervenir un terme supplémentaire de pénalisation de saut de flux qui vient améliorer les résultats. Les simulations numériques effectuées sur des cas tests (microprocesseur et radiateur en 2D, bloc métallique avec cartouches chauffantes en 3D) montrent la pertinence de la méthode. / This study concerns the simulation of thermal systems with multiple fields with a reduced-order method suited to a single body. Reduced models are constructed from modal branch eigenmodes. They are determined using a detailed model and allow the inclusion of non-linearities. The reduction is carried out by the simplified amalgam method which allows an automatic and immediate way to obtain a reduced basis which contains only a small number of modes from the original basis. The coupling between the different areas is carried out through a thermal contact resistance. It shows the need to involve an additional flux jump penalty term to improve the results. The numerical simulations carried out on test cases (microprocessor and radiator in 2D, metal block with hot cartridges in 3D) show the relevance of the method.

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